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Figur e 1.10. Snell's law

Exercise 1.13. Deduce Snell's law from the Fermat p

To describe optical properties of the medium, o

"unit sphere" S(X) at every point X: it consists of th

vectors at X. The hypersurface S is called indicatrix;

smooth, centrally symmetric and strictly convex. For e

case of Euclidean space, the indicatrices at all points

unit spheres. A field of indicatrices determines the so

metric: the distance between points A and B is the lea

light to get from A to B. A particular case of Finsler g

Riemannian one. In the latter case, one has a (varia

structure in the tangent space at every point X, and

S(X) is the unit sphere in this Euclidean structure.

Another example is a Minkowski metric. This is a

in a vector space whose indicatrices at different point

from each other by parallel translations. The speed

Minkowski space depends on the direction but not th

a homogeneous but anisotropic medium. Minkowski's

the study of these geometries came from number theo

Propagation of light satisfies the Huygens principl

A and consider the locus of points Ft reached by light

t. The hypersurface Ft is called a wave front, and it

points at Finsler distance t from A. The Huygens p

that the front Ft+£ can be constructed as follows: ever

lrThere was a heated polemic between Fermat and Descartes c

the speed of light increases or decreases with the density of the m

erroneously thought that light moves faster in water than in the air